Maximal torus

Examples

The unitary group U(n) has as a maximal torus the subgroup of all diagonal matrices. That is, : T = \left{\operatorname{diag}\left(e^{i\theta_1},e^{i\theta_2},\dots,e^{i\theta_n}\right) : \forall j, \theta_j \in \mathbb{R}\right}. T is clearly isomorphic to the product of n circles, so the unitary group U(n) has rank n. A maximal torus in the special unitary group SU(n) ⊂ U(n) is just the intersection of T and SU(n) which is a torus of dimension n − 1.

A maximal torus in the special orthogonal group SO(2n) is given by the set of all simultaneous rotations in any fixed choice of n pairwise orthogonal planes (i.e., two dimensional vector spaces). Concretely, one maximal torus consists of all block-diagonal matrices with 2\times 2 diagonal blocks, where each diagonal block is a rotation matrix. This is also a maximal torus in the group SO(2n+1) where the action fixes the remaining direction. Thus both SO(2n) and SO(2n+1) have rank n. For example, in the rotation group SO(3) the maximal tori are given by rotations about a fixed axis.

The symplectic group Sp(n) has rank n. A maximal torus is given by the set of all diagonal matrices whose entries all lie in a fixed complex subalgebra of H.

Properties

Let G be a compact, connected Lie group and let \mathfrak g be the Lie algebra of G. The first main result is the torus theorem, which may be formulated as follows: :Torus theorem: If T is one fixed maximal torus in G, then every element of G is conjugate to an element of T. This theorem has the following consequences:

• All maximal tori in G are conjugate.
• All maximal tori have the same dimension, known as the rank of G.
• A maximal torus in G is a maximal abelian subgroup, but the converse need not hold.
• The maximal tori in G are exactly the Lie subgroups corresponding to the maximal abelian subalgebras of \mathfrak g (cf. Cartan subalgebra)
• Every element of G lies in some maximal torus; thus, the exponential map for G is surjective.
• If G has dimension n and rank r then n − r is even.

Root system

If T is a maximal torus in a compact Lie group G, one can define a root system as follows. The roots are the weights for the adjoint action of T on the complexified Lie algebra of G. To be more explicit, let \mathfrak t denote the Lie algebra of T, let \mathfrak g denote the Lie algebra of G, and let \mathfrak g_{\mathbb C}:=\mathfrak g\oplus i\mathfrak g denote the complexification of \mathfrak g. Then we say that an element \alpha\in\mathfrak t is a root for G relative to T if \alpha\neq 0 and there exists a nonzero X\in\mathfrak g_{\mathbb C} such that :\mathrm{Ad}_{e^H}(X)=e^{i\langle\alpha,H\rangle}X for all H\in\mathfrak t. Here \langle\cdot,\cdot\rangle is a fixed inner product on \mathfrak g that is invariant under the adjoint action of connected compact Lie groups.

The root system, as a subset of the Lie algebra \mathfrak t of T, has all the usual properties of a root system, except that the roots may not span \mathfrak t. The root system is a key tool in understanding the classification and representation theory of G.

Weyl group

Given a torus T (not necessarily maximal), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is, :W(T,G) := N_G(T)/C_G(T). Fix a maximal torus T = T_0 in G; then the corresponding Weyl group is called the Weyl group of G (it depends up to isomorphism on the choice of T).

The first two major results about the Weyl group are as follows.

• The centralizer of T in G is equal to T, so the Weyl group is equal to N(T)/T.
• The Weyl group is generated by reflections about the roots of the associated Lie algebra. Thus, the Weyl group of T is isomorphic to the Weyl group of the root system of the Lie algebra of G.

We now list some consequences of these main results.

• Two elements in T are conjugate if and only if they are conjugate by an element of W. That is, each conjugacy class of G intersects T in exactly one Weyl orbit. In fact, the space of conjugacy classes in G is homeomorphic to the orbit space T/W.
• The Weyl group acts by (outer) automorphisms on T (and its Lie algebra).
• The identity component of the normalizer of T is also equal to T. The Weyl group is therefore equal to the component group of N(T).
• The Weyl group is finite. The representation theory of G is essentially determined by T and W.

As an example, consider the case G=SU(n) with T being the diagonal subgroup of G. Then x\in G belongs to N(T) if and only if x maps each standard basis element e_i to a multiple of some other standard basis element e_j, that is, if and only if x permutes the standard basis elements, up to multiplication by some constants. The Weyl group in this case is then the permutation group on n elements.

Weyl integral formula

Suppose f is a continuous function on G. Then the integral over G of f with respect to the normalized Haar measure dg may be computed as follows: : \displaystyle{\int_G f(g), dg = |W|^{-1} \int_T |\Delta(t)|^2\int_{G/T}f\left(yty^{-1}\right),d[y], dt,} where d[y] is the normalized volume measure on the quotient manifold G/T and dt is the normalized Haar measure on T. Here Δ is given by the Weyl denominator formula and |W| is the order of the Weyl group. An important special case of this result occurs when f is a class function, that is, a function invariant under conjugation. In that case, we have : \displaystyle{\int_G f(g), dg = |W|^{-1} \int_T f(t) |\Delta(t)|^2, dt.} Consider as an example the case G=SU(2), with T being the diagonal subgroup. Then the Weyl integral formula for class functions takes the following explicit form:

: \displaystyle{\int_{SU(2)} f(g), dg = \frac{1}{2} \int_0^{2\pi} f\left(\mathrm{diag}\left(e^{i\theta}, e^{-i\theta}\right)\right), 4,\mathrm{sin}^2(\theta) , \frac{d\theta}{2\pi}.} Here |W|=2, the normalized Haar measure on T is \frac{d\theta}{2\pi}, and \mathrm{diag}\left(e^{i\theta}, e^{-i\theta}\right) denotes the diagonal matrix with diagonal entries e^{i\theta} and e^{-i\theta}.

References

References

1. {{harvnb. Hall. 2015 Theorem 11.2
2. {{harvnb. Hall. 2015 Lemma 11.12
3. {{harvnb. Hall. 2015 Theorem 11.9
4. {{harvnb. Hall. 2015 Theorem 11.36 and Exercise 11.5
5. {{harvnb. Hall. 2015 Proposition 11.7
6. {{harvnb. Hall. 2015 Section 11.7
7. {{harvnb. Hall. 2015 Theorem 11.36
8. {{harvnb. Hall. 2015 Theorem 11.36
9. {{harvnb. Hall. 2015 Theorem 11.39
10. {{harvnb. Hall. 2015 Theorem 11.30 and Proposition 12.24
11. {{harvnb. Hall. 2015 Example 11.33